Dolbeault cohomology of compact complex manifolds with an action of a complex Lie group

TL;DR

This paper proves that a complex Lie group acting by holomorphic isometries on a compact complex Hermitian manifold induces a trivial action on Dolbeault and Bott-Chern cohomologies, and applies this to Vaisman manifolds.

Contribution

It establishes the triviality of the induced cohomology action for such group actions and computes Dolbeault cohomology for Vaisman manifolds.

Findings

Induced action on Dolbeault cohomology is trivial.

Induced action on Bott-Chern cohomology is trivial.

Dolbeault cohomology of Vaisman manifolds is computed.

Abstract

Let $G$ be a complex Lie group acting on a compact complex Hermitian manifold $M$ by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott-Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.